Skulls in the Stars

In a previous optics basics post, we discussed challenges associated with trying to define the velocity of a localized wave or ‘pulse’ of light. Traditional measurements of the velocity of an object involve measuring how far Δd an object travels in a certain amount of time Δt; then the velocity is simply

velocity = distance/time = Δd/Δt.

But a wave is an extended disturbance, not definitely associated with any particular point in space, and so measuring Δd becomes tricky. If there is a definite feature of the wave (such as a peak), we can define the velocity by measuring how fast the peak moves. If the wave changes shape (i.e. the peak disappears), as happens when waves propagate in matter, it is not immediately clear how one defines wave velocity.

The answer, as discussed previously, seems to be to define a ‘group’ velocity: we can mathematically characterize the velocity of the overall wave signal by

,

where Δω is the range of temporal frequencies in the wave pulse and Δk is the range of spatial wavenumbers in the pulse. This measure seemed quite good: under most circumstances, the quantity was less than the vacuum speed of light c, and therefore didn’t violate Einstein’s relativity, and those cases where the group velocity was greater than c seemed to always involve a significant attenuation or distortion of the wave.

However, in 2000 researchers Wang, Kuzmich and Dogariu from the NEC Research Institute shocked the physics and optics community by demonstrating* that materials exist for which the group velocity is greater than c, sometimes much greater than c, and the pulse travels at this higher speed without any obvious distortion or attenuation. What was going on?

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